# UNIFORM POLYNOMIAL TRICHOTOMY OF EVOLUTION OPERATORS IN BANACH SPACES

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We study the existence of in nite-dimensional vector spaces in the sets of norm-attaining operators, multilinear forms, and polynomials. Our main result is that, for every set of permutations P of the set {1, . . . , n}, there exists a closed infinite-dimensional Banach subspace of the space of...

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Let X and Y be Banach spaces with Y a proper subspace of X which is embedded into X by a continuous embedding. Let L(X) be the space of bounded operators on X. Then we show that if T âˆŠ L(Y) admits an extension T âˆŠ L(X) with T (X) âŠ† Y, then T and T have common spectral properties.

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We present a general method for constructing operators without non-trivial invariant closed subsets on a large class of non-reflexive Banach spaces. In particular, our approach unifies and generalizes several constructions due to Read of operators without non-trivial invariant subspaces on the...